Bihar Board Class 10 · Maths · Matric exam 2027
Chapter 2: Polynomials — notes
The chapter’s 6 parts in the book’s order — the first idea of each and the points Medha Ma’am writes on the board — and the summary at the end. The full lesson and a check question for each part are in the app.
1. What is a Polynomial, and its Degree
Okay, so you already know a bit about polynomials from Class IX.
- • Degree = highest power of x in p(x)
- • Linear polynomial: degree 1, e.g. 2x – 3
- • Quadratic polynomial: ax2 + bx + c, a ≠ 0
- • Cubic polynomial: ax3 + bx2 + cx + d, a ≠ 0
2. Zero of a Polynomial
Okay, so now let's talk about what a "zero" of a polynomial is.
- • p(k) = value of p(x) at x = k
- • k is a zero if p(k) = 0
- • p(x) = x2 – 3x – 4: p(–1) = 0, p(4) = 0
- • Zero of ax + b = –b/a (Constant term ÷ Coefficient of x, with sign)
3. Geometrical Meaning of Zeroes
Okay, so why do zeroes matter so much?
- • Zeroes = x-coordinates where graph meets x-axis
- • Linear graph: straight line, 1 zero
- • Quadratic graph: parabola, at most 2 zeroes
- • Cubic graph: at most 3 zeroes
- • x3 – 4x has zeroes –2, 0, 2
4. Relationship between Zeroes and Coefficients (Quadratic)
Okay, so — here's something beautiful.
- • p(x) = 2x2 – 8x + 6 = 2(x–1)(x–3)
- • Zeroes: 1 and 3
- • Sum of zeroes = α+β = –b/a
- • Product of zeroes = αβ = c/a
- • Example: sum=–3, product = 2 → x2+3x+2
5. Relationship between Zeroes and Coefficients (Cubic)
A similar relationship holds for cubic polynomials too, but now there are three zeroes and three relationships.
- • Cubic: ax3+bx2+cx+d, zeroes α, β, γ
- • α+β+γ = –b/a
- • αβ+βγ+γα = c/a
- • αβγ = –d/a
- • Check with p(x) = 2x3–5x2–14x+8, zeroes 4,–2,1/2
6. Division Algorithm for Polynomials
Suppose you know one zero of a cubic polynomial but want to find the others — division helps here.
- • Dividend = Divisor × Quotient + Remainder
- • p(x) = g(x) × q(x) + r(x)
- • r(x) = 0 or degree r(x) < degree g(x)
- • Example: 2x²+3x+1 ÷ (x+2) → quotient 2x–1, remainder 3
- • x3–3x2–x+3 = (x–1)(x+1)(x–3), zeroes 1,–1,3
Summary
Okay, so let's recap everything we did in this chapter. We named polynomials by their degree. A linear one has degree 1. A quadratic one has degree 2, written as ax² + bx + c. And a cubic one has degree 3, written as ax³ + bx² + cx + d. Then, we talked about a zero of a polynomial p(x) — it's just a value k where p(k) = 0. We also saw what a zero means on a graph. It's the x-coordinate where the graph of y = p(x) crosses the x-axis. And here's something neat — a polynomial of degree n has at most n zeroes. We also found a relationship between the zeroes and the coefficients. For a quadratic, sum of zeroes = −b/a, and product = c/a. For a cubic, α + β + γ = −b/a, αβ + βγ + γα = c/a, and αβγ = −d/a. Finally, we learned the Division Algorithm — p(x) = g(x) × q(x) + r(x). This lets us divide one polynomial by another, and find all the zeroes once we already know a few. Now, tell me — if you know one zero of a cubic polynomial, how do you think that helps you find the rest?
AskMedha · Medha Ma’am — Bihar Board Class 10 Maths, Chapter 2. Written from the textbook, checked twice.
Questions parents and students ask
- How do I get the "Polynomials" notes as a PDF?
- Press "Save as PDF" — the phone’s print sheet opens; choose "Save as PDF" there. The PDF holds the notes only.
- Where are these notes written from?
- From the chapter’s own textbook text, part by part, in the book’s own terms; a second, independent model then checked every part against the book. The full lesson, a check question for each part and the practice are in the AskMedha app.
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